# Fluid Mechanics and Hydraulics

Pressure

Unit Pressure
$p = \dfrac{F}{A}$

Absolute Pressure, Gage Pressure, and Atmospheric Pressure
$p_{abs} = p_{gage} + p_{atm}$

Variations in Pressure
$p_2 = p_1 + \gamma h$

$h = \dfrac{p}{\gamma}$

Total Hydrostatic Force

$F_x = p_{cg} A$

$F_y = \gamma V$

$F = \sqrt{{F_x}^2 + {F_y}^2}$

Eccentricity
$e = \dfrac{I_g}{A\bar{y}}$

Gravity Dam

$R_x = \Sigma F_h$

$R_y = \Sigma F_v$

$x = \dfrac{RM - OM}{R_y}$

$FS_s = \dfrac{\mu R_y}{R_x} \gt 1.0$

$FS_o = \dfrac{RM}{OM} \gt 1.0$

Buoyancy

Buoyant Force
$BF = \gamma V_D$

Volume Displaced
$V_D = \dfrac{s_{body}}{s_{liquid}} V = \dfrac{\gamma_{body}}{\gamma_{liquid}} V$

Draft
$D = \dfrac{s_{body}}{s_{liquid}} H = \dfrac{\gamma_{body}}{\gamma_{liquid}} H$

Area Submerged
$A_s = \dfrac{s_{body}}{s_{liquid}} A = \dfrac{\gamma_{body}}{\gamma_{liquid}} A$

Stability of Floating Body

Metacentric height
$MG = MB_o \pm GB_o$

Value of MBo
$MB_o = \dfrac{vs}{V_D \sin \theta}$

Thin-walled Pressure Vessel

Tangential Stress
$\sigma_T = \dfrac{pD}{2t}$

Longitudinal Stress
$\sigma_L = \dfrac{pD}{4t}$

Spherical Shell
$\sigma_L = \dfrac{pD}{4t}$

Spacing of Hoops
$s = \dfrac{2\sigma_h \, A_h}{pD}$

Moving Vessels

Horizontal Motion
$\tan \theta = \dfrac{a}{g}$

Inclined Motion
$\tan \theta = \dfrac{a_H}{g \pm a_V}$

Vertical Motion
$p = \gamma h \left( 1 \pm \dfrac{a}{g} \right)$

Rotating Vessel
$\tan \theta = \dfrac{\omega^2 x}{g}$

$y = \dfrac{\omega^2 x^2}{2g}$

Fluid Flow

Discharge
$Q = vA$

$h = \dfrac{v^2}{2g}$

$H = \dfrac{v^2}{2g} + \dfrac{p}{\gamma} + z$
$P = Q \gamma H$